Economic Models - Convex Optimization

Topic 1
A Novel Method of Estimation
Under Co-Integration
Alexis Lazaridis
Aristotle University of Thessaloniki, Greece
1. Introduction
We start with the unrestricted VAR(p):
p∑
x i = δ + µt i + A j x i−j + zd i + w i (1)
j=1
where x ∈ E n (here, E denotes the Euclidean space), t i and d i are the
trend and dummy, respectively and w is the noise vector. The matrices of
coefficients A j , are defined on E n × E n . After estimation, we may compute
matrix from:
⎛ ⎞
p∑
= ⎝ A j
⎠ − I
(1a)
j=1
or, directly, from the corresponding Error CorrectionVAR (ECVAR), which
is presented later.Also, an estimate of matrix can be obtained by applying
OLS, according to the procedure suggested by Kleibergen and Van Dijk
(1994). It is re-called that this type of VAR models like the one presented
above, have been recommended by Sims (1980), as a way to estimate the
dynamic relationships among jointly endogenous variables.
It is known that matrix can be decomposed such that = AC,
where the elements of A are the coefficients of adjustment and the rows of
the matrix C are the (possible) co-integrating vectors. It should be clarified
at the very outset, that in many books etc., C is denoted by β ′ (prime shows
transposition) and A by α, although capital letters are used for denoting
matrices. Besides, in the general linear model, y = Xβ + u,β is the vector
of coefficients. To avoid any confusion, we adopted the notation used here.
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